A Bargaining Model of Holdouts and Takings
by
Thomas J. Miceli
Professor of Economics
University of Connecticut
Storrs, CT 06269
Ph: (860) 486-5810
Fax: (860) 486-4463
e-mail: Thomas.Miceli@uconn.edu
Kathleen Segerson
Professor of Economics
University of Connecticut
Storrs, CT 06269
Ph: (860) 486-4567
Fax: (860) 486-4463
e-mail: Kathleen.Segerson@uconn.edu
April 2006
Preliminary: Do not cite or quote without permission
A Bargaining Model of Holdouts and Takings
1. Introduction
The holdout problem is commonly (and properly) cited as the justification for
forced sales of land under the government’s power of eminent domain. The threat of
holdouts arises when the government or a private developer attempts to assemble several
contiguous parcels of land in order to complete a large-scale development project like a
highway, railroad, or shopping center. In this setting, individual landowners, knowing
that each of their parcels is essential for completion of the overall project, may seek
prices in excess of their true valuations, thereby causing delay, increased costs, and
possibly failure to complete the project at all.
Though the holdout problem has been much discussed by scholars, few have
attempted to model it formally.1 As a result, its exact nature and the resulting
inefficiency are unclear. Some have described it as a problem of monopoly,2 while others
have characterized it in terms of transaction costs or breakdowns in bargaining.3 The
monopoly argument seems to suggest that projects involving holdouts will be
underprovided (due to the overpricing of land), while the bargaining cost approach tends
to focus on delay as the primary source of inefficiency.4
In this paper, we adopt the latter perspective and examine the holdout problem in
a bargaining framework. Specifically, we consider a developer or government agency
(henceforth, simply the developer) that needs to assemble several parcels of land for a
1
Two exceptions are Strange (1995) and Menezes and Pitchford (2001).
See, for example, Posner (2005, p. 55) and Knetsch and Borcherding (1979, p. 244).
3
See Shavell (2005, p. 125) and Cooter and Ulen (1988, p. 193).
4
On the latter point, see Fischel (1995, p. 68) and Hirsch (1999, p. 28).
2
1
large-scale project. The developer can negotiate individually, or jointly, with the owners
and can do so in one of two periods. If, at the end of the second period, the developer has
acquired all of the parcels, he can proceed with the project, but if not (i.e., if he has only
acquired some or none of the parcels), he will scrap it. To capture the cost of delay, we
assume that the gross value of the project is higher if it can be completed after the first
period rather than after the second. We assume that individual sellers behave noncooperatively in deciding when to come to the bargaining table, but that any sellers at the
table in a given period engage in a Nash bargaining game with the developer to determine
the prices of their individual parcels.5
In this setting, we show that holding out (i.e., waiting until the second period to
bargain) is always a Nash equilibrium, and in some cases it is the only equilibrium. This
is true even though completion of the project in the first period is a Pareto-superior
outcome. The reason for the inefficiency is that each seller would gain privately by
waiting to sell in period two if all other sellers sell in period one.
Given this result, we ask what will happen if the developer is given the power to
use eminent domain in period two to take any parcels that he was not able to acquire
consensually in period one.6 If compensation for takings is set at a property’s market
value (which we assume is less than its true value to the owner), we show that the unique
equilibrium in the face of a takings threat is for all owners to sell in period one.
Intuitively, owners sell in period one to avoid the loss from a taking in period two. Thus,
the mere threat of eminent domain is sufficient to overcome the holdout problem.
5
In this respect, the model closely resembles that developed by Menezes and Pitchford (2001) in a different
context.
6
We actually show that it is always optimal for the developer to try to bargain in period one, even if he has
the right to use eminent domain in that period as well.
2
Although this outcome is potentially welfare-improving because it avoids the cost
of delay, we note two offsetting costs. First, because owners sell in the face of a takings
threat, they may sell at prices below their true reservation prices. As a result, the
developer is able to capture a larger share of the surplus from the project. This represents
a purely distributional effect, but it may nevertheless impose a social cost to the extent
that society cares about fairness and/or private property rights. A second cost of eminent
domain, however, is allocative, and also derives from the underpricing of the properties
relative to the owners’ opportunity costs. This is the possibility that developers may go
ahead with inefficient projects—that is, projects whose gross value is less than the sum of
the values of the individual parcels to their owners. This is possible in our model, even
though eminent domain is never actually used, because of the effect of its threatened use
on land prices.
We conclude the paper by discussing some related issues in the context of the
model, such as the proper measure of compensation for takings, whether the holdout
problem is more severe for public versus private land assemblers (a question that relates
to the definition of “public use”), and the extent to which eminent domain may be “selflimiting.”
2. Notation and Assumptions
Consider a developer who wishes to acquire two adjacent (and identical) parcels
of land, designated A and B, for a development project. Let
V = gross value of the development if completed;
v = value of each parcel to its owner (the owner’s reservation price);
3
m = market value of each parcel; also the value of a single parcel to the developer.
We assume that
V > 2v,
(1)
reflecting the complementariness of the parcels when consolidated for development.
Given (1), it is efficient for the developer to acquire the parcels. We also assume,
however, that
v > m,
(2)
reflecting the fact that owners typically value their land in excess of its market value.
This also implies that it is not efficient for the developer to acquire only one parcel.
We suppose that bargaining between the developer and the landowners can take
place at two dates: t=1 (now), and t=2 (later). The developer can thus proceed with the
development under any of the following scenarios: (a) he succeeds in buying both parcels
now, (b) he buys one now and one later, (c) and he buys both later. However, if he is
unable to acquire both parcels by t=2, we assume that he abandons the project altogether,
resulting in a return of zero if he has acquired neither parcel, and m if he has acquired
only one.
Note that scenarios (b) and (c) differ from (a) in that completion of the project is
delayed. We assume that this imposes a cost on the developer equal to δ>0. However,
we also assume that
V – δ > 2v,
(3)
4
implying that it is still efficient to complete the development at this date. The cost of
delay is the sole cost associated with the holdout problem in the current model.7 Finally,
we assume that all payoffs are realized at t=2, and there is no discounting.
3. Bargaining
We first consider the case where period two has arrived and the developer has
acquired neither of the parcels. That is, both owners were period-one holdouts. Since (3)
implies there is still a surplus to be divided among the three parties, and there is no
possible bargaining in the future, both sellers have an interest in engaging in bargaining
now.8 If the developer acquires both parcels, his net return is
V–δ–P2A–P2B,
(4)
where P2A and P2B are the prices obtained by owners A and B in period 2.9 As noted, the
developer’s threat point is zero since failure to acquire both parcels will result in the
project’s being scrapped. As for sellers, their return from a sale is simply the price, and
their threat points are v. Using the Nash bargaining solution applied to a three-way
bargain (Osborne and Rubinstein, 1990, p. 23), we obtain the prices by solving:
(V–δ–P2A–P2B )(P2A–v)(P2B–v).
max
A
B
(5)
P2 , P2
Since the sellers are identical, each sells for
V   v
Pˆ2 
.
3
(6)
This reflects Cohen’s (1991, p. 354) argument that buyers and sellers ultimately recognize their joint
interests in completing a sale. Another interpretation of δ is that it is the transaction costs the developer
must incur in order to strike a bargain with holdouts. Regardless of the interpretation, one would expect δ
to be increasing in the number of parcels that must be assembled, and also possibly in the magnitude of the
surplus generated by assembly.
8
In contrast, condition (2) implies that a unilateral sale would never be profitable in t=2.
9
Henceforth, superscripts denote parcels and subscripts denote periods.
7
5
The resulting net return to the developer, found by setting P2A=P2B= P̂2 in (4), is
V    2v
,
3
(7)
which is positive given (3).
Consider next the case where the developer succeeded in acquiring parcel A in
period 1 for a price P1 A, and now seeks to buy parcel B in period two for a price P2B. If
he succeeds his net return is again given by (4), but his threat point is now m– P1 A. The
situation for the seller is the same as before. Thus, P2B solves
[(V–δ– P1 A–P2B ) – (m– P1 A)](P2B–v).
max
B
(8)
P2
The result is
P2 * 
V  v   m
.
2
(9)
(Note that this price is independent of the period one price.) Comparison of (6) and (9)
shows that P2*> P̂2 given (2) and (3). Thus, it is better for a seller to be a lone holdout
compared to the case where both holdout and sell in period two.
Now move back to period one. There are three possibilities here: (a) both sellers
bargain (and sell), (b) one sells and one holds out, and (c) both holdout. We have already
seen that case (c) results in sale by both sellers in period to for P̂2 . Thus, we need only
consider (a) and (b). In case (a), both sellers bargain. If both sell, the developer’s net
return is V–P1A–P1B,10 and each seller receives the sale price. As for the threat points,
note that a failure to reach a bargain in period one still allows the parties to bargain in
period two. We assume that if one seller walks away, they both do, but then both bargain
10
Note that there is no cost of delay in this case.
6
in period two, as described above. Thus, the threat points for all three parties are the net
returns from that bargain (specifically, the price in (6) for each seller and the return in (7)
for the developer). Bringing all of this together, we can determine the period one prices
from a three-way bargain as the solution to

 V    2v   A  V    v   B  V    v 
A
B
(
V

P

P
)

max

  P1  
  P1  
 . (10)
1
1

P1A , P1B
3
3
3

 

 



Symmetry again implies that both sellers sell for the same price:
V v
Pˆ1 
.
3
(11)
Comparison of (6) and (11) shows that Pˆ1  Pˆ2 . That is, sellers would prefer to
sell jointly in period one rather than period two, owing to the larger surplus available to
be divided in period one. Further, comparison of (9) and (11) shows that P2*

P̂1 . Thus,

a seller may or may not prefer being the lone holdout in period one compared to the case
where both sell in one. The reason for the ambiguity here is the cost of delay, δ, which
favors a sale by both in period one. Thus, the larger is δ the more likely it is that
P2*< P̂1 .11
The final case is where owner A sells in period one while owner B holds out.
Both the developer and the willing seller anticipate that the holdout will sell in period two
for P2* (assuming that the period-one sale occurs), and the threat points, as above, are the
returns from a joint sale in period two. Thus, P1A in the current case is determined by the
solution to
11
The specific condition for P2* to be larger is 2δ<V–δ+v–3m, where the right-hand side is positive by (2)
and (3). Thus, P2* will definitely be larger if δ=0.
7

 V    2v   A  V    v 
max
(V    P1A  P2 *)  
  P1  
 ,
A 
P1
3
3

 



(12)
and the result is
P1 * 
V  v   m
.
4
(13)
Using our previous results, we can show that P2*>P1*, and Pˆ1  Pˆ2  P1 * . Thus, being
the lone seller in period one is clearly the worst possible outcome.
4. Equilibrium Behavior of Sellers
Now that we have derived the equilibrium prices under the assumption of
cooperative bargaining among those at the bargaining table, we turn to the choices of
individual sellers regarding when to come to the bargaining table and when to holdout.
In examining this choice, we assume that the sellers behave non-cooperatively.
In the first period, each seller has two possible strategies: bargain, or holdout.
The analysis in the previous section determined the prices that would result from each
possible combination of these strategies, and they are shown in the following table:
Seller 2
Bargain
Holdout
Bargain
Pˆ1 , Pˆ1
P1*,P2*
Holdout
P2*,P1*
Pˆ2 , Pˆ2
Seller 1
8
(Note that because we assumed that any period-one holdouts would bargain in period
two, there is no separate decision in that period. Thus, the strategy “holdout” means
holdout now and bargain in period two.)
The nature of the equilibria depend on relationship between P2* and P̂1 , which we
showed above is ambiguous. Thus, there are two possibilities. First, when P2*> P̂1 , there
is a unique Nash equilibrium wherein both sellers holdout. In this case, the individual
gain from being the lone holdout dominates the joint benefits from a period-one sale by
both sellers. Thus, holding out is a dominant strategy for both sellers. Alternatively,
when P2*< P̂1 , there are two pure strategy Nash equilibria: (bargain, bargain) and
(holdout, holdout). In this case, the best strategy depends on what each seller thinks the
other will do. Although both sellers are better off if they jointly bargain (since Pˆ1  Pˆ2 ),
each wants to guard against the possibility of being the lone seller in period one.
In terms of efficiency, the social return (equal to the sum of the returns of the
three parties) is V when both sellers agree to sell in period one, and V–δ when both
holdout in period one and then sell in period two. This reflects the social cost associated
with delay. Despite this cost, the results in this section show that there is no way to
guarantee that a holdout problem will not occur, and in some cases, it is the only
equilibrium.
5. Bargaining in the Shadow of Eminent Domain
This section asks whether the threat of a taking by the developer can help to
alleviate the risk of a holdout problem. We assume that the developer first seeks to
negotiate a consensual sale with both sellers in period one (as above), but that he has
9
been granted the right to use eminent domain in period two to take any properties that he
was unable to acquire in period one. (We show below that it is always optimal for the
developer to try to negotiate a sale before resorting to eminent domain anyway.) If the
developer takes a property, he has to pay the owner its market value, m, and he also
incurs a transaction/litigation cost, τ, per property taken. We assume sellers incur no
transaction costs when their land is taken, but they do incur a loss given m<v.
As above, we begin in period two by considering the case where neither property
was acquired in period one (i.e., both sellers held out). In this case, the developer will
take both properties provided that
V – δ – 2(m+τ) ≥ 0.
(14)
Note that a sufficient condition for (14) to hold, given (3), is m+τ≤v. Similarly, if the
developer was only able to acquire one property in period one for a price P1, he will take
the second in period two provided that
V – δ – (m+τ) – P1 ≥ m – P1,
or
V – δ – 2m – τ ≥ 0,
(15)
which is satisfied given (14).
Assuming that (14) holds, the threat to use eminent domain in period two is
credible. Thus, we turn to the bargaining between the developer and the sellers in period
one in the shadow of this threat. Consider first the case where both sellers bargain. In
this case, the threat point for the sellers is m, while the threat point for the developer is
the left-hand side of (14). The prices therefore solve


max
(V  P1A  P1B )  (V    2(m   ) ( P1A  m)( P1B  m) ,
A
B
P1 , P1
10
(16)
which yields
2  
Pˆ1  m 
.
3
(17)
Sellers are thus able to extract a price in excess of their market value because, by selling
in period one, they save the developer the cost of delay plus the transaction costs of using
eminent domain. Nevertheless, there is no guarantee that (17) is larger than the sellers’
reservation price, v. This reflects the developer’s credible threat to take the land for m in
period two. (We return to this point in the next section.) Finally, note that the developer
is better off seeking to bargain in period one rather than resorting to eminent domain
immediately since his surplus from doing to (the term in square brackets in (16)) turns
out to be (δ+2τ)/3 in equilibrium, which is clearly positive.
Consider next the case where only seller A bargains in period one while seller B
holds out. The threat points are the same as in (16), but in this case, the developer’s
return if he succeeds in buying parcel A is V–δ–P1A–(m+τ) since he will have to take
parcel B by eminent domain in period two. The sale price for parcel A, P1A, is therefore
determined by
[(V–δ–P1A–(m+τ))–(V–δ–2(m+τ))](P1A–m),
max
A
(18)
P1
and the resulting price is
P1* = m +

.
2
(19)
Comparison of (17) and (19) shows that Pˆ1  P1 *>m. Thus, it is best for both sellers to
sell in period one, next best to be the lone seller in period one, and worst to be a holdout.
Given these results, we now reconsider the equilibrium decisions of sellers in the
face of eminent domain. The payoff matrix in this case is given by
11
Seller 2
Bargain
Holdout
Bargain
Pˆ1 , Pˆ1
P1*, m
Holdout
m , P1*
m, m
Seller 1
where P̂1 and P1* are now given by (17) and (19). The unique Nash equilibrium is for
both sellers to bargain. Thus, the threat of eminent domain does in fact succeed in
eliminating the holdout problem. (Note that because eminent domain is never actually
used in equilibrium, the litigation costs are never incurred.)
6. The Costs of Eminent Domain
The preceding analysis has shown that the economic benefit of eminent domain is
avoidance of the holdout problem. In this sense, it improves efficiency by saving the cost
of delay. In this section, however, we note two offsetting costs of eminent domain, both
of which arise from the possibility that owners, when faced with the threat of a taking,
may sell their properties for less than their opportunity costs. First, note that the
condition for this underpricing to occur is that the expression in (17) is less than v, or
v–m>
2  
,
3
(20)
which is more likely to hold the larger is the owner’s “subjective value,” v–m, and the
smaller are the developer’s litigation costs and costs of delay.
When (20) holds, the threat of eminent domain redistributes the gains from
development in favor of the developer compared to a world in which sellers have the
right to refuse any offer. While this does not represent a loss in efficiency when
12
condition (1) holds (i.e., given that the project promises a net benefit), it does entail a loss
in welfare to the extent that considerations of fairness and/or respect for private property
rights matter to society.12
In addition, however, the possibility that owners will agree to sell their properties
for less than their reservation prices may entail a loss in efficiency due to the risk of
excessive takings. The problem is that there may be no way to ensure that the targeted
land is more valuable to the developer than to the owners absent the owners’ right to
refuse to sell. In ordinary market transactions, an owners’ right of refusal guarantees that
any transfer of property is value-enhancing in the sense that the buyer values the property
more than the seller. In contrast, when the developer has the power of eminent domain,
we have shown that owners may consent to sell for prices below their true reservation
price for fear that their land will be taken for an even smaller amount of compensation
later. This will result in inefficient projects going forward if 2v>V>2 P̂1 , or if v> P̂1 ,
which is the same as condition (20). The crux of the problem, of course, is the
unobservability of v, which forces courts to set compensation at the market value of the
property rather than its true value to the owner (see the discussion in the next section).
The preceding illustrates the fundamental trade-off associated with eminent
domain. On one hand, there is the risk of costly delay in the production of large scale
projects due to the holdout problem, but on the other there is the fairness cost plus the
possibility of overinvestment in such projects due to the underpricing of properties in the
This reflects Epstein’s (1985, Chapter 12) concern about the allocation of the surplus from a taking as
between the beneficiaries of the taking and those whose land is taken.
12
13
face of a takings threat.13 The optimal scope of eminent domain must balance these
offsetting factors.
7. Related Issues
This section discusses several related issues in light of the model.
The measure of compensation. We noted above that both the fairness and
efficiency costs of eminent domain arose from the underpricing of properties relative to
their true value to the owners. The underlying problem is that the market value of a piece
of property (m) is typically less than its value to the owner (v) owing to what Fischel
(1995, pp. 207-209) has called the offer-ask disparity.14 If v were observable,
compensation could simply be set at that amount, thereby eliminating both the fairness
and efficiency costs of eminent domain.15
A possible response to this undervaluation problem when v is not observable is to
pay owners a set premium above their market value as a way of compensating them for
their subjective value (Epstein, 1985, pp. 174-175). To be sure, the amount of this
premium would be somewhat arbitrary, but one possibility would be to tie it to the length
of time that the owner (or his ancestors) has occupied the property, in recognition of the
probable link between subjective value and length of possession.16 For example,
compensation could be set at m(1+π(t)), where the premium π is increasing in time of
possession, t.
13
This same trade-off arises whenever a liability rule is substituted for a property rule. See generally
Calabresi and Melamed (1972) and Kaplow and Shavell (1996).
14
Also see Knetsch and Borcherding (1979).
15
This is not entirely true, since owners would still be deprived of the right to negotiate for a share of the
surplus from the transaction, given by the difference V–2v.
16
As Oliver Wendell Holmes (1897, p. 477) observed, “A thing which you have enjoyed and used as your
own for a long time, whether property or an opinion, takes root in your being and cannot be torn away
without your resenting the act and trying to defend yourself…”
14
Public use. The question of public use arises in cases where a private entity, as
opposed to the government, is granted the power of eminent domain. While courts have
typically approved of this,17 it remains a controversial question. Since the holdout
problem is the proper economic justification for eminent domain,18 the key question is
whether this problem poses a greater threat to public versus private land assemblers. The
model in this paper suggested no such distinction, but some have argued that holdouts
may be less of a concern for private assemblers because they can more easily conceal
their intent through the use of buying agents, option agreements, and other devices, while
the government must operate in the open (Merrill, 1986, p. 82; Fischel, 1995, p. 70).
This argument has some plausibility, but it is not entirely convincing since even private
developers generally have to obtain various permits before proceeding with most large
scale projects, thereby bringing their intentions into the open.
Another possible distinction between private and public takings concerns the
magnitude of the public benefits. When the project in question is a public good, the
benefits are, by definition, widely spread, which may justify the relatively small costs that
the use of eminent domain imposes on a few landowners. (Of course, eminent domain is
only justified, even for provision of public goods, if land assembly is necessary.) In
contrast, when the benefits of a project are largely private, society may be less willing to
impose any uncompensated costs on individual owners, even if a holdout problem exists.
Under this interpretation, only those private projects that promise widespread public
(external) benefits would satisfy the “public use” test.
17
18
The most recent such decision is the Supreme Court’s ruling in Kelo v. City of New London (2005).
See Merrill’s (1986) discussion of the “means-ends” distinction. Also see Epstein (1985, Chapter 12).
15
Eminent domain is self-limiting. As a final point, note that the left- and righthand sides of (20) may not be independent. Specifically, as a landowner’s subjective
value, v–m, grows, he is likely to fight harder to avoid a taking, in which case the
litigation cost of using eminent domain, τ, will tend to increase as well. Based on this
logic, Merrill (1986, p. 101) and Fischel (1995, p. 74) have argued that eminent domain
will tend to be “self-limiting,” especially in cases where the cost of holdouts (as captured
by δ on the right-hand side of (20)) is small.
16
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17